Octave 3.8, jcobi/2

Percentage Accurate: 62.9% → 97.4%

Time: 20.7s

Alternatives: 10

Speedup: 4.8×

Specification

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0) 2.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Initial Program: 62.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0) 2.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Alternative 1: 97.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.5:\\ \;\;\;\;\frac{2 \cdot i + 0.5 \cdot \left(2 + \beta \cdot 2\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.5)
     (/ (+ (* 2.0 i) (* 0.5 (+ 2.0 (* beta 2.0)))) alpha)
     (/
      (+
       (/
        (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
        (+ alpha (+ beta (fma 2.0 i 2.0))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5) {
		tmp = ((2.0 * i) + (0.5 * (2.0 + (beta * 2.0)))) / alpha;
	} else {
		tmp = ((((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / (alpha + (beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.5)
		tmp = Float64(Float64(Float64(2.0 * i) + Float64(0.5 * Float64(2.0 + Float64(beta * 2.0)))) / alpha);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[(2.0 * i), $MachinePrecision] + N[(0.5 * N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.5

   1. Initial program 2.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l/ 2.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]

      2. associate-+l+ 2.0%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]

      3. +-commutative 2.0%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i\right)} + 1}{2} \]

      4. associate-+l+ 2.0%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \color{blue}{\left(\beta + \left(\alpha + 2 \cdot i\right)\right)}} + 1}{2} \]

   3. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around inf 93.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}} \]
   6. Step-by-step derivation

      1. associate-*r/ 93.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)}{\alpha}} \]

      2. *-commutative 93.9%

         \[\leadsto \frac{\color{blue}{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right) \cdot 0.5}}{\alpha} \]

      3. distribute-rgt1-in 93.9%

         \[\leadsto \frac{\left(\color{blue}{\left(-1 + 1\right) \cdot \beta} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right) \cdot 0.5}{\alpha} \]

      4. metadata-eval 93.9%

         \[\leadsto \frac{\left(\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right) \cdot 0.5}{\alpha} \]

      5. mul-1-neg 93.9%

         \[\leadsto \frac{\left(0 \cdot \beta - \color{blue}{\left(-\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)}\right) \cdot 0.5}{\alpha} \]

      6. fma-define 93.9%

         \[\leadsto \frac{\left(0 \cdot \beta - \left(-\left(2 + \color{blue}{\mathsf{fma}\left(2, \beta, 4 \cdot i\right)}\right)\right)\right) \cdot 0.5}{\alpha} \]

      7. *-commutative 93.9%

         \[\leadsto \frac{\left(0 \cdot \beta - \left(-\left(2 + \mathsf{fma}\left(2, \beta, \color{blue}{i \cdot 4}\right)\right)\right)\right) \cdot 0.5}{\alpha} \]

   7. Simplified 93.9%

      \[\leadsto \color{blue}{\frac{\left(0 \cdot \beta - \left(-\left(2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)\right)\right)\right) \cdot 0.5}{\alpha}} \]

   8. Taylor expanded in i around 0 93.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} + 2 \cdot \frac{i}{\alpha}} \]

   9. Taylor expanded in alpha around 0 93.9%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(2 + 2 \cdot \beta\right) + 2 \cdot i}{\alpha}} \]

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 81.8%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{2 \cdot i + 0.5 \cdot \left(2 + \beta \cdot 2\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{t\_0}{t\_1}}{2 + t\_1}\\ \mathbf{if}\;t\_2 \leq -0.5:\\ \;\;\;\;\frac{2 \cdot i + 0.5 \cdot \left(2 + \beta \cdot 2\right)}{\alpha}\\ \mathbf{elif}\;t\_2 \leq 0.0002:\\ \;\;\;\;\frac{1 + \frac{t\_0}{\left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* (+ alpha beta) (- beta alpha)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (/ (/ t_0 t_1) (+ 2.0 t_1))))
   (if (<= t_2 -0.5)
     (/ (+ (* 2.0 i) (* 0.5 (+ 2.0 (* beta 2.0)))) alpha)
     (if (<= t_2 0.0002)
       (/
        (+
         1.0
         (/
          t_0
          (*
           (+ (+ alpha beta) (+ 2.0 (* 2.0 i)))
           (+ beta (+ alpha (* 2.0 i))))))
        2.0)
       (/ (+ 1.0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) * (beta - alpha);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (t_0 / t_1) / (2.0 + t_1);
	double tmp;
	if (t_2 <= -0.5) {
		tmp = ((2.0 * i) + (0.5 * (2.0 + (beta * 2.0)))) / alpha;
	} else if (t_2 <= 0.0002) {
		tmp = (1.0 + (t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i)))))) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (alpha + beta) * (beta - alpha)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = (t_0 / t_1) / (2.0d0 + t_1)
    if (t_2 <= (-0.5d0)) then
        tmp = ((2.0d0 * i) + (0.5d0 * (2.0d0 + (beta * 2.0d0)))) / alpha
    else if (t_2 <= 0.0002d0) then
        tmp = (1.0d0 + (t_0 / (((alpha + beta) + (2.0d0 + (2.0d0 * i))) * (beta + (alpha + (2.0d0 * i)))))) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) / (beta + (alpha + 2.0d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) * (beta - alpha);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (t_0 / t_1) / (2.0 + t_1);
	double tmp;
	if (t_2 <= -0.5) {
		tmp = ((2.0 * i) + (0.5 * (2.0 + (beta * 2.0)))) / alpha;
	} else if (t_2 <= 0.0002) {
		tmp = (1.0 + (t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i)))))) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) * (beta - alpha)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = (t_0 / t_1) / (2.0 + t_1)
	tmp = 0
	if t_2 <= -0.5:
		tmp = ((2.0 * i) + (0.5 * (2.0 + (beta * 2.0)))) / alpha
	elif t_2 <= 0.0002:
		tmp = (1.0 + (t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i)))))) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) * Float64(beta - alpha))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(Float64(t_0 / t_1) / Float64(2.0 + t_1))
	tmp = 0.0
	if (t_2 <= -0.5)
		tmp = Float64(Float64(Float64(2.0 * i) + Float64(0.5 * Float64(2.0 + Float64(beta * 2.0)))) / alpha);
	elseif (t_2 <= 0.0002)
		tmp = Float64(Float64(1.0 + Float64(t_0 / Float64(Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i))) * Float64(beta + Float64(alpha + Float64(2.0 * i)))))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) * (beta - alpha);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = (t_0 / t_1) / (2.0 + t_1);
	tmp = 0.0;
	if (t_2 <= -0.5)
		tmp = ((2.0 * i) + (0.5 * (2.0 + (beta * 2.0)))) / alpha;
	elseif (t_2 <= 0.0002)
		tmp = (1.0 + (t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i)))))) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 / t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.5], N[(N[(N[(2.0 * i), $MachinePrecision] + N[(0.5 * N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$2, 0.0002], N[(N[(1.0 + N[(t$95$0 / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta + N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.5

   1. Initial program 2.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l/ 2.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]

      2. associate-+l+ 2.0%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]

      3. +-commutative 2.0%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i\right)} + 1}{2} \]

      4. associate-+l+ 2.0%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \color{blue}{\left(\beta + \left(\alpha + 2 \cdot i\right)\right)}} + 1}{2} \]

   3. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around inf 93.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}} \]
   6. Step-by-step derivation

      1. associate-*r/ 93.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)}{\alpha}} \]

      2. *-commutative 93.9%

         \[\leadsto \frac{\color{blue}{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right) \cdot 0.5}}{\alpha} \]

      3. distribute-rgt1-in 93.9%

         \[\leadsto \frac{\left(\color{blue}{\left(-1 + 1\right) \cdot \beta} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right) \cdot 0.5}{\alpha} \]

      4. metadata-eval 93.9%

         \[\leadsto \frac{\left(\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right) \cdot 0.5}{\alpha} \]

      5. mul-1-neg 93.9%

         \[\leadsto \frac{\left(0 \cdot \beta - \color{blue}{\left(-\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)}\right) \cdot 0.5}{\alpha} \]

      6. fma-define 93.9%

         \[\leadsto \frac{\left(0 \cdot \beta - \left(-\left(2 + \color{blue}{\mathsf{fma}\left(2, \beta, 4 \cdot i\right)}\right)\right)\right) \cdot 0.5}{\alpha} \]

      7. *-commutative 93.9%

         \[\leadsto \frac{\left(0 \cdot \beta - \left(-\left(2 + \mathsf{fma}\left(2, \beta, \color{blue}{i \cdot 4}\right)\right)\right)\right) \cdot 0.5}{\alpha} \]

   7. Simplified 93.9%

      \[\leadsto \color{blue}{\frac{\left(0 \cdot \beta - \left(-\left(2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)\right)\right)\right) \cdot 0.5}{\alpha}} \]

   8. Taylor expanded in i around 0 93.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} + 2 \cdot \frac{i}{\alpha}} \]

   9. Taylor expanded in alpha around 0 93.9%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(2 + 2 \cdot \beta\right) + 2 \cdot i}{\alpha}} \]

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 2.0000000000000001e-4

   1. Initial program 100.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l/ 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]

      2. associate-+l+ 100.0%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i\right)} + 1}{2} \]

      4. associate-+l+ 100.0%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \color{blue}{\left(\beta + \left(\alpha + 2 \cdot i\right)\right)}} + 1}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   if 2.0000000000000001e-4 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 45.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 91.7%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   6. Step-by-step derivation

      1. associate-+r+ 91.7%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}} + 1}{2} \]

      2. +-commutative 91.7%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta} + 1}{2} \]

   7. Simplified 91.7%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}} + 1}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{2 \cdot i + 0.5 \cdot \left(2 + \beta \cdot 2\right)}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.0002:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 + i \cdot 4}{\alpha}\\ t_1 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 1950000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\alpha \leq 1.15 \cdot 10^{+58}:\\ \;\;\;\;\frac{t\_0}{2}\\ \mathbf{elif}\;\alpha \leq 3.5 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_0 + \frac{\beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ 2.0 (* i 4.0)) alpha))
        (t_1 (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))
   (if (<= alpha 1950000000000.0)
     t_1
     (if (<= alpha 1.15e+58)
       (/ t_0 2.0)
       (if (<= alpha 3.5e+153) t_1 (+ (* 0.5 t_0) (/ beta alpha)))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (2.0 + (i * 4.0)) / alpha;
	double t_1 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 1950000000000.0) {
		tmp = t_1;
	} else if (alpha <= 1.15e+58) {
		tmp = t_0 / 2.0;
	} else if (alpha <= 3.5e+153) {
		tmp = t_1;
	} else {
		tmp = (0.5 * t_0) + (beta / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (2.0d0 + (i * 4.0d0)) / alpha
    t_1 = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    if (alpha <= 1950000000000.0d0) then
        tmp = t_1
    else if (alpha <= 1.15d+58) then
        tmp = t_0 / 2.0d0
    else if (alpha <= 3.5d+153) then
        tmp = t_1
    else
        tmp = (0.5d0 * t_0) + (beta / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (2.0 + (i * 4.0)) / alpha;
	double t_1 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 1950000000000.0) {
		tmp = t_1;
	} else if (alpha <= 1.15e+58) {
		tmp = t_0 / 2.0;
	} else if (alpha <= 3.5e+153) {
		tmp = t_1;
	} else {
		tmp = (0.5 * t_0) + (beta / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (2.0 + (i * 4.0)) / alpha
	t_1 = (1.0 + (beta / (beta + 2.0))) / 2.0
	tmp = 0
	if alpha <= 1950000000000.0:
		tmp = t_1
	elif alpha <= 1.15e+58:
		tmp = t_0 / 2.0
	elif alpha <= 3.5e+153:
		tmp = t_1
	else:
		tmp = (0.5 * t_0) + (beta / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(2.0 + Float64(i * 4.0)) / alpha)
	t_1 = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0)
	tmp = 0.0
	if (alpha <= 1950000000000.0)
		tmp = t_1;
	elseif (alpha <= 1.15e+58)
		tmp = Float64(t_0 / 2.0);
	elseif (alpha <= 3.5e+153)
		tmp = t_1;
	else
		tmp = Float64(Float64(0.5 * t_0) + Float64(beta / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (2.0 + (i * 4.0)) / alpha;
	t_1 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	tmp = 0.0;
	if (alpha <= 1950000000000.0)
		tmp = t_1;
	elseif (alpha <= 1.15e+58)
		tmp = t_0 / 2.0;
	elseif (alpha <= 3.5e+153)
		tmp = t_1;
	else
		tmp = (0.5 * t_0) + (beta / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 1950000000000.0], t$95$1, If[LessEqual[alpha, 1.15e+58], N[(t$95$0 / 2.0), $MachinePrecision], If[LessEqual[alpha, 3.5e+153], t$95$1, N[(N[(0.5 * t$95$0), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.95e12 or 1.15000000000000001e58 < alpha < 3.4999999999999999e153

   1. Initial program 82.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 84.6%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   6. Step-by-step derivation

      1. associate-+r+ 84.6%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}} + 1}{2} \]

      2. +-commutative 84.6%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta} + 1}{2} \]

   7. Simplified 84.6%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}} + 1}{2} \]

   8. Taylor expanded in alpha around 0 89.3%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]

   if 1.95e12 < alpha < 1.15000000000000001e58

   1. Initial program 26.9%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 34.1%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around inf 71.6%

      \[\leadsto \frac{\color{blue}{\frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 71.8%

      \[\leadsto \frac{\color{blue}{\frac{2 + 4 \cdot i}{\alpha}}}{2} \]
   7. Step-by-step derivation

      1. *-commutative 71.8%

         \[\leadsto \frac{\frac{2 + \color{blue}{i \cdot 4}}{\alpha}}{2} \]

   8. Simplified 71.8%

      \[\leadsto \frac{\color{blue}{\frac{2 + i \cdot 4}{\alpha}}}{2} \]

   if 3.4999999999999999e153 < alpha

   1. Initial program 1.4%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 26.4%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around inf 80.8%

      \[\leadsto \frac{\color{blue}{\frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 80.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 4 \cdot i}{\alpha} + \frac{\beta}{\alpha}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1950000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.15 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 3.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{2 + i \cdot 4}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 1950000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq 5.7 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 3.5 \cdot 10^{+153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot i + 0.5 \cdot \left(2 + \beta \cdot 2\right)}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))
   (if (<= alpha 1950000000000.0)
     t_0
     (if (<= alpha 5.7e+57)
       (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)
       (if (<= alpha 3.5e+153)
         t_0
         (/ (+ (* 2.0 i) (* 0.5 (+ 2.0 (* beta 2.0)))) alpha))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 1950000000000.0) {
		tmp = t_0;
	} else if (alpha <= 5.7e+57) {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	} else if (alpha <= 3.5e+153) {
		tmp = t_0;
	} else {
		tmp = ((2.0 * i) + (0.5 * (2.0 + (beta * 2.0)))) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    if (alpha <= 1950000000000.0d0) then
        tmp = t_0
    else if (alpha <= 5.7d+57) then
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    else if (alpha <= 3.5d+153) then
        tmp = t_0
    else
        tmp = ((2.0d0 * i) + (0.5d0 * (2.0d0 + (beta * 2.0d0)))) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 1950000000000.0) {
		tmp = t_0;
	} else if (alpha <= 5.7e+57) {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	} else if (alpha <= 3.5e+153) {
		tmp = t_0;
	} else {
		tmp = ((2.0 * i) + (0.5 * (2.0 + (beta * 2.0)))) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0
	tmp = 0
	if alpha <= 1950000000000.0:
		tmp = t_0
	elif alpha <= 5.7e+57:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	elif alpha <= 3.5e+153:
		tmp = t_0
	else:
		tmp = ((2.0 * i) + (0.5 * (2.0 + (beta * 2.0)))) / alpha
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0)
	tmp = 0.0
	if (alpha <= 1950000000000.0)
		tmp = t_0;
	elseif (alpha <= 5.7e+57)
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	elseif (alpha <= 3.5e+153)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(2.0 * i) + Float64(0.5 * Float64(2.0 + Float64(beta * 2.0)))) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	tmp = 0.0;
	if (alpha <= 1950000000000.0)
		tmp = t_0;
	elseif (alpha <= 5.7e+57)
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	elseif (alpha <= 3.5e+153)
		tmp = t_0;
	else
		tmp = ((2.0 * i) + (0.5 * (2.0 + (beta * 2.0)))) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 1950000000000.0], t$95$0, If[LessEqual[alpha, 5.7e+57], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 3.5e+153], t$95$0, N[(N[(N[(2.0 * i), $MachinePrecision] + N[(0.5 * N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.95e12 or 5.6999999999999998e57 < alpha < 3.4999999999999999e153

   1. Initial program 82.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 84.6%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   6. Step-by-step derivation

      1. associate-+r+ 84.6%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}} + 1}{2} \]

      2. +-commutative 84.6%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta} + 1}{2} \]

   7. Simplified 84.6%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}} + 1}{2} \]

   8. Taylor expanded in alpha around 0 89.3%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]

   if 1.95e12 < alpha < 5.6999999999999998e57

   1. Initial program 26.9%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 34.1%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around inf 71.6%

      \[\leadsto \frac{\color{blue}{\frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 71.8%

      \[\leadsto \frac{\color{blue}{\frac{2 + 4 \cdot i}{\alpha}}}{2} \]
   7. Step-by-step derivation

      1. *-commutative 71.8%

         \[\leadsto \frac{\frac{2 + \color{blue}{i \cdot 4}}{\alpha}}{2} \]

   8. Simplified 71.8%

      \[\leadsto \frac{\color{blue}{\frac{2 + i \cdot 4}{\alpha}}}{2} \]

   if 3.4999999999999999e153 < alpha

   1. Initial program 1.4%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l/ 0.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]

      2. associate-+l+ 0.0%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]

      3. +-commutative 0.0%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i\right)} + 1}{2} \]

      4. associate-+l+ 0.0%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \color{blue}{\left(\beta + \left(\alpha + 2 \cdot i\right)\right)}} + 1}{2} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around inf 80.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}} \]
   6. Step-by-step derivation

      1. associate-*r/ 80.8%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)}{\alpha}} \]

      2. *-commutative 80.8%

         \[\leadsto \frac{\color{blue}{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right) \cdot 0.5}}{\alpha} \]

      3. distribute-rgt1-in 80.8%

         \[\leadsto \frac{\left(\color{blue}{\left(-1 + 1\right) \cdot \beta} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right) \cdot 0.5}{\alpha} \]

      4. metadata-eval 80.8%

         \[\leadsto \frac{\left(\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right) \cdot 0.5}{\alpha} \]

      5. mul-1-neg 80.8%

         \[\leadsto \frac{\left(0 \cdot \beta - \color{blue}{\left(-\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)}\right) \cdot 0.5}{\alpha} \]

      6. fma-define 80.8%

         \[\leadsto \frac{\left(0 \cdot \beta - \left(-\left(2 + \color{blue}{\mathsf{fma}\left(2, \beta, 4 \cdot i\right)}\right)\right)\right) \cdot 0.5}{\alpha} \]

      7. *-commutative 80.8%

         \[\leadsto \frac{\left(0 \cdot \beta - \left(-\left(2 + \mathsf{fma}\left(2, \beta, \color{blue}{i \cdot 4}\right)\right)\right)\right) \cdot 0.5}{\alpha} \]

   7. Simplified 80.8%

      \[\leadsto \color{blue}{\frac{\left(0 \cdot \beta - \left(-\left(2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)\right)\right)\right) \cdot 0.5}{\alpha}} \]

   8. Taylor expanded in i around 0 80.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} + 2 \cdot \frac{i}{\alpha}} \]

   9. Taylor expanded in alpha around 0 80.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(2 + 2 \cdot \beta\right) + 2 \cdot i}{\alpha}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1950000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 5.7 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 3.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot i + 0.5 \cdot \left(2 + \beta \cdot 2\right)}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1700000000000 \lor \neg \left(\alpha \leq 2.15 \cdot 10^{+58}\right) \land \alpha \leq 1.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (or (<= alpha 1700000000000.0)
         (and (not (<= alpha 2.15e+58)) (<= alpha 1.5e+168)))
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if ((alpha <= 1700000000000.0) || (!(alpha <= 2.15e+58) && (alpha <= 1.5e+168))) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((alpha <= 1700000000000.0d0) .or. (.not. (alpha <= 2.15d+58)) .and. (alpha <= 1.5d+168)) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if ((alpha <= 1700000000000.0) || (!(alpha <= 2.15e+58) && (alpha <= 1.5e+168))) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if (alpha <= 1700000000000.0) or (not (alpha <= 2.15e+58) and (alpha <= 1.5e+168)):
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if ((alpha <= 1700000000000.0) || (!(alpha <= 2.15e+58) && (alpha <= 1.5e+168)))
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if ((alpha <= 1700000000000.0) || (~((alpha <= 2.15e+58)) && (alpha <= 1.5e+168)))
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[Or[LessEqual[alpha, 1700000000000.0], And[N[Not[LessEqual[alpha, 2.15e+58]], $MachinePrecision], LessEqual[alpha, 1.5e+168]]], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.7e12 or 2.14999999999999996e58 < alpha < 1.4999999999999999e168

   1. Initial program 81.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 97.2%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 84.3%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   6. Step-by-step derivation

      1. associate-+r+ 84.3%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}} + 1}{2} \]

      2. +-commutative 84.3%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta} + 1}{2} \]

   7. Simplified 84.3%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}} + 1}{2} \]

   8. Taylor expanded in alpha around 0 88.9%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]

   if 1.7e12 < alpha < 2.14999999999999996e58 or 1.4999999999999999e168 < alpha

   1. Initial program 8.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 27.5%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around inf 79.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 74.6%

      \[\leadsto \frac{\color{blue}{\frac{2 + 4 \cdot i}{\alpha}}}{2} \]
   7. Step-by-step derivation

      1. *-commutative 74.6%

         \[\leadsto \frac{\frac{2 + \color{blue}{i \cdot 4}}{\alpha}}{2} \]

   8. Simplified 74.6%

      \[\leadsto \frac{\color{blue}{\frac{2 + i \cdot 4}{\alpha}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1700000000000 \lor \neg \left(\alpha \leq 2.15 \cdot 10^{+58}\right) \land \alpha \leq 1.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 1950000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq 4 \cdot 10^{+58}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\alpha \leq 1.7 \cdot 10^{+168}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))
   (if (<= alpha 1950000000000.0)
     t_0
     (if (<= alpha 4e+58)
       (/ 1.0 alpha)
       (if (<= alpha 1.7e+168) t_0 (/ (+ beta 1.0) alpha))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 1950000000000.0) {
		tmp = t_0;
	} else if (alpha <= 4e+58) {
		tmp = 1.0 / alpha;
	} else if (alpha <= 1.7e+168) {
		tmp = t_0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    if (alpha <= 1950000000000.0d0) then
        tmp = t_0
    else if (alpha <= 4d+58) then
        tmp = 1.0d0 / alpha
    else if (alpha <= 1.7d+168) then
        tmp = t_0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 1950000000000.0) {
		tmp = t_0;
	} else if (alpha <= 4e+58) {
		tmp = 1.0 / alpha;
	} else if (alpha <= 1.7e+168) {
		tmp = t_0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0
	tmp = 0
	if alpha <= 1950000000000.0:
		tmp = t_0
	elif alpha <= 4e+58:
		tmp = 1.0 / alpha
	elif alpha <= 1.7e+168:
		tmp = t_0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0)
	tmp = 0.0
	if (alpha <= 1950000000000.0)
		tmp = t_0;
	elseif (alpha <= 4e+58)
		tmp = Float64(1.0 / alpha);
	elseif (alpha <= 1.7e+168)
		tmp = t_0;
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	tmp = 0.0;
	if (alpha <= 1950000000000.0)
		tmp = t_0;
	elseif (alpha <= 4e+58)
		tmp = 1.0 / alpha;
	elseif (alpha <= 1.7e+168)
		tmp = t_0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 1950000000000.0], t$95$0, If[LessEqual[alpha, 4e+58], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[alpha, 1.7e+168], t$95$0, N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.95e12 or 3.99999999999999978e58 < alpha < 1.70000000000000001e168

   1. Initial program 81.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 97.2%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 84.3%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   6. Step-by-step derivation

      1. associate-+r+ 84.3%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}} + 1}{2} \]

      2. +-commutative 84.3%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta} + 1}{2} \]

   7. Simplified 84.3%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}} + 1}{2} \]

   8. Taylor expanded in alpha around 0 88.9%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]

   if 1.95e12 < alpha < 3.99999999999999978e58

   1. Initial program 26.9%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 34.1%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around inf 71.6%

      \[\leadsto \frac{\color{blue}{\frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 71.8%

      \[\leadsto \frac{\color{blue}{\frac{2 + 4 \cdot i}{\alpha}}}{2} \]
   7. Step-by-step derivation

      1. *-commutative 71.8%

         \[\leadsto \frac{\frac{2 + \color{blue}{i \cdot 4}}{\alpha}}{2} \]

   8. Simplified 71.8%

      \[\leadsto \frac{\color{blue}{\frac{2 + i \cdot 4}{\alpha}}}{2} \]

   9. Taylor expanded in i around 0 66.0%

      \[\leadsto \color{blue}{\frac{1}{\alpha}} \]

   if 1.70000000000000001e168 < alpha

   1. Initial program 1.3%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 25.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 14.7%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   6. Step-by-step derivation

      1. associate-+r+ 14.7%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}} + 1}{2} \]

      2. +-commutative 14.7%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta} + 1}{2} \]

   7. Simplified 14.7%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}} + 1}{2} \]

   8. Taylor expanded in alpha around inf 50.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]
   9. Step-by-step derivation

      1. associate-*r/ 50.8%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(2 + 2 \cdot \beta\right)}{\alpha}} \]

      2. distribute-rgt-in 50.8%

         \[\leadsto \frac{\color{blue}{2 \cdot 0.5 + \left(2 \cdot \beta\right) \cdot 0.5}}{\alpha} \]

      3. metadata-eval 50.8%

         \[\leadsto \frac{\color{blue}{1} + \left(2 \cdot \beta\right) \cdot 0.5}{\alpha} \]

      4. *-commutative 50.8%

         \[\leadsto \frac{1 + \color{blue}{\left(\beta \cdot 2\right)} \cdot 0.5}{\alpha} \]

      5. associate-*l* 50.8%

         \[\leadsto \frac{1 + \color{blue}{\beta \cdot \left(2 \cdot 0.5\right)}}{\alpha} \]

      6. metadata-eval 50.8%

         \[\leadsto \frac{1 + \beta \cdot \color{blue}{1}}{\alpha} \]

   10. Simplified 50.8%

       \[\leadsto \color{blue}{\frac{1 + \beta \cdot 1}{\alpha}} \]

   11. Taylor expanded in alpha around 0 50.8%

       \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha}} \]
   12. Step-by-step derivation

       1. +-commutative 50.8%

          \[\leadsto \frac{\color{blue}{\beta + 1}}{\alpha} \]

   13. Simplified 50.8%

       \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1950000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 4 \cdot 10^{+58}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\alpha \leq 1.7 \cdot 10^{+168}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1950000000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 5.5 \cdot 10^{+57} \lor \neg \left(\alpha \leq 1.22 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1950000000000.0)
   0.5
   (if (or (<= alpha 5.5e+57) (not (<= alpha 1.22e+168))) (/ 1.0 alpha) 0.5)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1950000000000.0) {
		tmp = 0.5;
	} else if ((alpha <= 5.5e+57) || !(alpha <= 1.22e+168)) {
		tmp = 1.0 / alpha;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1950000000000.0d0) then
        tmp = 0.5d0
    else if ((alpha <= 5.5d+57) .or. (.not. (alpha <= 1.22d+168))) then
        tmp = 1.0d0 / alpha
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1950000000000.0) {
		tmp = 0.5;
	} else if ((alpha <= 5.5e+57) || !(alpha <= 1.22e+168)) {
		tmp = 1.0 / alpha;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1950000000000.0:
		tmp = 0.5
	elif (alpha <= 5.5e+57) or not (alpha <= 1.22e+168):
		tmp = 1.0 / alpha
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1950000000000.0)
		tmp = 0.5;
	elseif ((alpha <= 5.5e+57) || !(alpha <= 1.22e+168))
		tmp = Float64(1.0 / alpha);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1950000000000.0)
		tmp = 0.5;
	elseif ((alpha <= 5.5e+57) || ~((alpha <= 1.22e+168)))
		tmp = 1.0 / alpha;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1950000000000.0], 0.5, If[Or[LessEqual[alpha, 5.5e+57], N[Not[LessEqual[alpha, 1.22e+168]], $MachinePrecision]], N[(1.0 / alpha), $MachinePrecision], 0.5]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.95e12 or 5.5000000000000002e57 < alpha < 1.21999999999999991e168

   1. Initial program 81.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l/ 80.7%

         \[\leadsto \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]

      2. associate-+l+ 80.7%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]

      3. +-commutative 80.7%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i\right)} + 1}{2} \]

      4. associate-+l+ 80.7%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \color{blue}{\left(\beta + \left(\alpha + 2 \cdot i\right)\right)}} + 1}{2} \]

   3. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 73.2%

      \[\leadsto \color{blue}{0.5} \]

   if 1.95e12 < alpha < 5.5000000000000002e57 or 1.21999999999999991e168 < alpha

   1. Initial program 8.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 27.5%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around inf 79.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 74.6%

      \[\leadsto \frac{\color{blue}{\frac{2 + 4 \cdot i}{\alpha}}}{2} \]
   7. Step-by-step derivation

      1. *-commutative 74.6%

         \[\leadsto \frac{\frac{2 + \color{blue}{i \cdot 4}}{\alpha}}{2} \]

   8. Simplified 74.6%

      \[\leadsto \frac{\color{blue}{\frac{2 + i \cdot 4}{\alpha}}}{2} \]

   9. Taylor expanded in i around 0 50.6%

      \[\leadsto \color{blue}{\frac{1}{\alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1950000000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 5.5 \cdot 10^{+57} \lor \neg \left(\alpha \leq 1.22 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 450000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 450000.0) 0.5 (/ (- 2.0 (/ 2.0 beta)) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 450000.0) {
		tmp = 0.5;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 450000.0d0) then
        tmp = 0.5d0
    else
        tmp = (2.0d0 - (2.0d0 / beta)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 450000.0) {
		tmp = 0.5;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 450000.0:
		tmp = 0.5
	else:
		tmp = (2.0 - (2.0 / beta)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 450000.0)
		tmp = 0.5;
	else
		tmp = Float64(Float64(2.0 - Float64(2.0 / beta)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 450000.0)
		tmp = 0.5;
	else
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 450000.0], 0.5, N[(N[(2.0 - N[(2.0 / beta), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5e5

   1. Initial program 75.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.6%

         \[\leadsto \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]

      2. associate-+l+ 75.6%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]

      3. +-commutative 75.6%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i\right)} + 1}{2} \]

      4. associate-+l+ 75.6%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \color{blue}{\left(\beta + \left(\alpha + 2 \cdot i\right)\right)}} + 1}{2} \]

   3. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 76.0%

      \[\leadsto \color{blue}{0.5} \]

   if 4.5e5 < beta

   1. Initial program 49.1%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 94.3%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 75.5%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   6. Step-by-step derivation

      1. associate-+r+ 75.5%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}} + 1}{2} \]

      2. +-commutative 75.5%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta} + 1}{2} \]

   7. Simplified 75.5%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}} + 1}{2} \]

   8. Taylor expanded in alpha around 0 75.6%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]

   9. Taylor expanded in beta around inf 74.9%

      \[\leadsto \frac{\color{blue}{2 - 2 \cdot \frac{1}{\beta}}}{2} \]
   10. Step-by-step derivation

       1. associate-*r/ 74.9%

          \[\leadsto \frac{2 - \color{blue}{\frac{2 \cdot 1}{\beta}}}{2} \]

       2. metadata-eval 74.9%

          \[\leadsto \frac{2 - \frac{\color{blue}{2}}{\beta}}{2} \]

   11. Simplified 74.9%

       \[\leadsto \frac{\color{blue}{2 - \frac{2}{\beta}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 450000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3 \cdot 10^{+23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 5.3e+23) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.3e+23) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5.3d+23) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.3e+23) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.3e+23:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.3e+23)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.3e+23)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 5.3e+23], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 5.3000000000000001e23

   1. Initial program 76.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l/ 76.4%

         \[\leadsto \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]

      2. associate-+l+ 76.4%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]

      3. +-commutative 76.4%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i\right)} + 1}{2} \]

      4. associate-+l+ 76.4%

         \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \color{blue}{\left(\beta + \left(\alpha + 2 \cdot i\right)\right)}} + 1}{2} \]

   3. Simplified 76.4%

      \[\leadsto \color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 75.1%

      \[\leadsto \color{blue}{0.5} \]

   if 5.3000000000000001e23 < beta

   1. Initial program 45.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 93.9%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf 76.9%

      \[\leadsto \frac{\color{blue}{2}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.3 \cdot 10^{+23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.3% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.5)

C

double code(double alpha, double beta, double i) {
	return 0.5;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.5d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.5;
}

Python

def code(alpha, beta, i):
	return 0.5

Julia

function code(alpha, beta, i)
	return 0.5
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.5;
end

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 67.0%

   \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
2. Step-by-step derivation

   1. associate-/l/ 66.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]

   2. associate-+l+ 66.4%

      \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]

   3. +-commutative 66.4%

      \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i\right)} + 1}{2} \]

   4. associate-+l+ 66.4%

      \[\leadsto \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \color{blue}{\left(\beta + \left(\alpha + 2 \cdot i\right)\right)}} + 1}{2} \]

3. Simplified 66.4%

   \[\leadsto \color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}} \]
4. Add Preprocessing

5. Taylor expanded in i around inf 62.8%

   \[\leadsto \color{blue}{0.5} \]

6. Final simplification 62.8%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))